Subgroups — #507092

Order 512.

317 conjugacy classes of subgroups.

The subgroup lattice has 185 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#29230	C₂	1	2	yes	maximal
2	#13513	C₂	1	4	yes	maximal
3	#13188	C₂	1	2	yes	maximal
4	#13173	C₂	1	2	yes	maximal
5	#12911	C₂	1	2	yes	maximal
6	#12909	C₂	1	1	yes	maximal, characteristic
7	#12670	C₂	1	1	yes	maximal, characteristic
8	#12664	C₂	1	1	yes	maximal, characteristic
9	C₂×#368	C₂²	1	2	yes	—
10	C₂×#368	C₂²	1	1	yes	characteristic
11	C₂×#368	C₂²	1	2	yes	—
12	#2158	C₂²	1	1	yes	characteristic
13	#2155	C₂²	1	4	yes	—
14	C₂×#364	C₂²	1	1	yes	characteristic
15	#2108	C₂²	1	2	yes	—
16	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
17	#2102	—	2	4	no	—
18	#1953	—	2	2	no	—
19	#1953	C₂²	1	4	yes	—
20	#1848	C₂²	1	2	yes	—
21	#1833	—	2	2	no	—
22	#1833	C₂²	1	4	yes	—
23	#1772	C₂²	1	2	yes	—
24	#1772	C₂²	1	1	yes	characteristic
25	#1388	C₂²	1	4	yes	—
26	#1382	C₂²	1	2	yes	—
27	#1379	C₂²	1	2	yes	—
28	C₂×M₃₂	C₂³	1	2	yes	—
29	C₂²×C₁₆	C₂³	1	1	yes	characteristic
30	C₂²×C₁₆	C₂³	1	1	yes	characteristic
31	#430	—	2	4	no	—
32	C₂×#108	—	2	2	no	—
33	C₂×#108	D₄	1	1	yes	characteristic
34	C₂×#108	C₂³	1	4	yes	—
35	C₂×#108	C₂³	1	2	yes	—
36	#427	D₄	1	1	yes	characteristic
37	#427	C₂³	1	2	yes	—
38	#423	—	2	2	no	—
39	#405	—	2	2	no	—
40	#405	C₂³	1	2	yes	—
41	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
42	#368	—	2	1	no	—
43	#368	—	2	1	no	—
44	#368	—	2	2	no	—
45	#368	—	2	2	no	—
46	#368	—	4	2	no	—
47	#367	—	2	1	no	—
48	#366	—	2	1	no	—
49	#365	—	4	4	no	—
50	#364	Q₈	1	2	yes	—
51	#364	Q₈	1	2	yes	—
52	#346	—	2	2	no	—
53	C₄×C₁₆	—	2	2	no	—
54	C₄×C₁₆	Q₈	1	1	yes	characteristic
55	C₄×C₁₆	Q₈	1	1	yes	characteristic
56	C₄×C₁₆	Q₈	1	1	yes	characteristic
57	C₄×C₁₆	Q₈	1	1	yes	characteristic
58	#344	—	2	4	no	—
59	#344	—	2	4	no	—
60	#339	—	2	2	no	—
61	#339	D₄	1	2	yes	—
62	H(3,ℤ/4)	—	2	4	no	—
63	C₂×M₁₆	—	2	4	no	—
64	C₂×M₁₆	—	2	2	no	—
65	C₂×M₁₆	C₂×D₄	1	2	yes	—
66	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₈	C₂⁴	1	1	yes	characteristic
68	#117	—	2	2	no	—
69	#117	C₂×D₄	1	2	yes	—
70	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
71	M₃₂	—	2	2	no	—
72	M₃₂	C₂×Q₈	1	4	yes	—
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	—	2	1	no	—
75	C₂×C₁₆	—	2	1	no	—
76	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₁₆	#41	1	1	yes	characteristic
81	C₂×C₁₆	#41	1	1	yes	characteristic
82	C₂×C₁₆	#41	1	1	yes	characteristic
83	C₂×C₁₆	#41	1	1	yes	characteristic
84	#108	—	2	1	no	—
85	#108	—	2	1	no	—
86	#108	—	2	2	no	—
87	#108	—	2	4	no	—
88	#108	—	2	4	no	—
89	#108	—	2	2	no	—
90	#108	—	4	2	no	—
91	#108	—	4	2	no	—
92	#108	—	8	4	no	—
93	#107	—	2	1	no	—
94	#107	—	2	2	no	—
95	#106	—	2	1	no	—
96	#106	—	2	2	no	—
97	C₄⋊C₈	—	8	2	no	—
98	#97	—	2	2	no	—
99	#97	—	4	2	no	—
100	C₄×C₈	—	2	2	no	—
101	C₄×C₈	—	4	2	no	—
102	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
103	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
104	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
105	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	2	2	no	—
110	M₁₆	—	4	2	no	—
111	M₁₆	—	4	4	no	—
112	M₁₆	—	4	4	no	—
113	M₁₆	—	4	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	4	4	no	—
117	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
119	C₂×C₈	#135	1	1	yes	characteristic
120	C₂×C₈	#135	1	1	yes	characteristic
121	C₂×C₈	#135	1	1	yes	characteristic
122	C₂×C₈	#135	1	1	yes	characteristic
123	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₈	C₂×#41	1	1	yes	characteristic
126	C₂×C₈	C₂×#41	1	1	yes	characteristic
127	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₂²⋊C₄	—	2	2	no	—
132	C₂²⋊C₄	—	4	2	no	—
133	C₄²	—	2	1	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	4	2	no	—
136	C₄²	C₄⋊Q₈	1	1	yes	characteristic
137	C₄²	C₄⋊Q₈	1	1	yes	characteristic
138	C₁₆	—	2	1	no	—
139	C₁₆	—	2	1	no	—
140	C₁₆	—	2	4	no	—
141	C₁₆	#125	1	1	yes	characteristic
142	C₁₆	#125	1	1	yes	characteristic
143	C₁₆	#125	1	1	yes	characteristic
144	C₁₆	#125	1	1	yes	characteristic
145	C₂³	#578	1	1	yes	characteristic
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
152	C₂×C₄	#547	1	1	yes	characteristic
153	C₂×C₄	#550	1	1	yes	characteristic
154	C₂×C₄	C₂×#135	1	1	yes	characteristic
155	C₂×C₄	C₂×#135	1	1	yes	characteristic
156	C₂×C₄	#576	1	1	yes	characteristic
157	C₈	—	4	2	no	—
158	C₈	—	8	2	no	—
159	C₈	—	8	4	no	—
160	C₈	#499	1	1	yes	characteristic
161	C₈	#499	1	1	yes	characteristic
162	C₈	#499	1	1	yes	characteristic
163	C₈	#499	1	1	yes	characteristic
164	C₈	C₂×#125	1	1	yes	characteristic
165	C₈	#544	1	1	yes	characteristic
166	C₈	#554	1	1	yes	characteristic
167	C₈	#557	1	1	yes	characteristic
168	C₂²	—	4	1	no	—
169	C₂²	#3164	1	1	yes	characteristic
170	C₂²	#3342	1	1	yes	characteristic
171	C₂²	#3393	1	1	yes	characteristic
172	C₄	—	2	2	no	—
173	C₄	—	4	1	no	—
174	C₄	—	8	2	no	—
175	C₄	—	16	2	no	—
176	C₄	#2552	1	1	yes	characteristic
177	C₄	C₂×#499	1	1	yes	characteristic
178	C₄	#3341	1	1	yes	characteristic
179	C₄	#3396	1	1	yes	characteristic
180	C₂	—	2	1	no	—
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	#28828	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#3341
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#28828
prime radical	C₂³	#578
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#29230	2
#13513	2
#13188	2
#13173	2
#12911	2
#12909	2
#12670	2
#12664	2